Theorem 3.55 Rudin (rearrangement and convergence)

The sum $s_n$ comprises $a_1,\, \dotsc,\, a_N$, and also $a_{N+1},\, \dotsc,\, a_n$. The sum $s_n'$ comprises $a_1,\, \dotsc,\, a_N$, and also the $a_k$ for $k$ in a finite set $F$ disjoint from $\{1,\, \dotsc,\, N\}$. Let $G = \{N+1,\, \dotsc,\, n\}$. Then

$$s_n - s_n' = \sum_{k\in G} a_k - \sum_{k\in F} a_k = \sum_{k\in G\setminus F} a _k - \sum_{k \in F\setminus G} a_k,$$

whence

$$\lvert s_n - s_n'\rvert \leqslant \sum_{k \in G\setminus F} \lvert a_k\rvert + \sum_{k \in F\setminus G} \lvert a_k\rvert = \sum_{k \in (G\setminus F) \cup (F\setminus G)} \lvert a_k\rvert \leqslant \varepsilon,$$

since $(G\setminus F) \cup (F\setminus G) \subset \{N+1,\, \dotsc,\, m\}$ for large enough $m$.